Abstract
In [G1], [G2], the distribution of modular symbols is studied and a new class of functions which satisfy a transformation law involving these objects is introduced. The goal of Goldfeld’s program is to prove Szpiro’s conjecture which states that for elliptic curves with minimal discriminant D and conductor N there is an absolute constant κ such that D ? N. To do this, an equivalent conjecture involving modular symbols is established in [G4] using the, now proven, conjecture of Shimura, Taniyama and Weil. It seems that a sufficiently good understanding of the new series proposed by Goldfeld should yield a resolution of these conjectures. We repeat here their definition in a somewhat more general form to include period polynomials rather than modular symbols only. For positive integers M,N such that M |N, let m, k be non-negative integers such that m ≥ k − 2 ≥ 0 and let χ be a Dirichlet character modulo N. First, for each f ∈ Sk(M) = {holomorphic cusp forms of weight k and level M} we denote by rf the map which sends γ ∈ Γ0(M), the Hecke congruence group of level M , to the polynomial function:
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